*2.1.2 Second case* μrz <0

For *εr<sup>θ</sup>* >0, *εTE <sup>r</sup>*,*eff* is rewritten as

$$\varepsilon\_{r,\theta\overline{f}}^{TE} = |\ \varepsilon\_{r\theta}| \left( 1 + \frac{1}{|\ \varepsilon\_{r\theta}\,\,\mu\_{rz}|k\_0^2} \cdot \left(\frac{u\_{nm}'}{R}\right)^2 \right) > 0. \tag{32}$$

*H*ð Þ*<sup>e</sup>*

with

guide.

*H*ð Þ*<sup>e</sup>*

*<sup>r</sup>* <sup>¼</sup> �*jωε*<sup>0</sup> *Kc:<sup>θ</sup>:Kc:<sup>r</sup>*

> *<sup>θ</sup>* <sup>¼</sup> �*jωε*<sup>0</sup> *Kc:<sup>r</sup>*

*Jn* of the first kind of order n.

Eq. (42) gives:

with

**71**

ffiffiffiffiffiffi *εrθ* p ffiffiffiffiffi *<sup>ε</sup>rr* <sup>p</sup> *: n r E*0*:*sin

*DOI: http://dx.doi.org/10.5772/intechopen.91645*

ffiffiffiffiffi *<sup>ε</sup>rr* <sup>p</sup> ffiffiffiffiffi

*PTM* <sup>¼</sup>

ð *R* 2 ð*π*

0

*N*ð Þ*<sup>e</sup>*

8 < :

Finally, the propagation constant in TM mode is given by:

*δ<sup>n</sup>* ¼

*k*ð Þ *TM <sup>z</sup>:nm* ¼ �

> *f* ð Þ *TM <sup>c</sup>:nm* <sup>¼</sup> *<sup>c</sup>* 2*π*

to describe the propagation characteristics of the waveguide modes.

Obviously, the cutoff frequency is written

*E*ð Þ*<sup>e</sup> <sup>r</sup> <sup>H</sup>*<sup>∗</sup> ð Þ*<sup>e</sup>*

*<sup>E</sup>*<sup>0</sup> <sup>¼</sup> *<sup>K</sup>*<sup>2</sup>

*nm* <sup>¼</sup> <sup>1</sup> *unm:J* 0 *<sup>n</sup>*ð Þ *unm :*

2*π*, *if n* ¼ 0 *<sup>π</sup>* � sin 4ð Þ *<sup>π</sup>b:*<sup>n</sup>

*<sup>b</sup>* <sup>¼</sup> *<sup>K</sup>*ð Þ*<sup>e</sup>*

*k*2

*K*ð Þ*<sup>e</sup> <sup>c</sup>:<sup>r</sup>:* ffiffiffiffiffiffi *εrθ*

0

*<sup>ε</sup>rz* <sup>p</sup> *:E*0*:* cos

The boundary condition (18) gives the following equation

*unm* ¼

*K*ð Þ*<sup>e</sup> <sup>c</sup>:<sup>θ</sup>:* ffiffiffiffiffi *εrr* p

*Rigorous Analysis of the Propagation in Metallic Circular Waveguide with Discontinuities…*

!

*n:θ*

*J* 0 *n*

*n:θ*

*Jn*

ffiffiffiffiffi *εrz* p ffiffiffiffiffi *<sup>ε</sup>rr* <sup>p</sup> *<sup>K</sup>*ð Þ*<sup>e</sup> <sup>c</sup>:r:r* � �*<sup>e</sup>*

ffiffiffiffiffi *εrz* p ffiffiffiffiffi *<sup>ε</sup>rr* <sup>p</sup> *<sup>K</sup>*ð Þ*<sup>e</sup> <sup>c</sup>:r:r* � �*<sup>e</sup>*

*Jn*ð Þ¼ *unm* 0*:* (40)

*<sup>c</sup>:r:R:* (41)

*nm* (43)

<sup>q</sup> (44)

<sup>p</sup> (46)

� �*:* (48)

(45)

(47)

�*jkzz* (38)

�*jkzz* (39)

*K*ð Þ*<sup>e</sup> <sup>c</sup>:r:* ffiffiffiffiffiffi *εrθ* p

!

ffiffiffiffiffi *εrz* p ffiffiffiffiffi *<sup>ε</sup>rr* <sup>p</sup> *<sup>K</sup>*ð Þ*<sup>e</sup>*

In Eq. (41) *unm* represents the mth zero (m = 1, 2, 3, … ) of the Bessel function

The constant *E*<sup>0</sup> is determined by normalizing the power flow down the circular

*<sup>θ</sup>* � *<sup>E</sup>*ð Þ*<sup>e</sup>*

*c:r* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *ωε*0*εrrkz* <sup>p</sup> *<sup>N</sup>*ð Þ*<sup>e</sup>*

*<sup>θ</sup> H*<sup>∗</sup> ð Þ*<sup>e</sup> <sup>r</sup>*

� �*rdrd<sup>θ</sup>* <sup>¼</sup> <sup>1</sup> (42)

ffiffiffi *δn* 2

4b*:*<sup>n</sup> , *if n*<sup>&</sup>gt; <sup>0</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

� �<sup>2</sup> r

*εrz*

*R*

*unm R*

*<sup>c</sup>:<sup>θ</sup>:* ffiffiffiffiffi *εrr* p

<sup>0</sup>*εrr:μr<sup>θ</sup>* � *<sup>ε</sup>rr*

1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>μ</sup>r<sup>θ</sup>* j j *<sup>ε</sup>rz* <sup>p</sup> *: unm*

We can introduce the following effective permeability and effective permittivity

*K*ð Þ*<sup>e</sup> <sup>c</sup>:<sup>θ</sup>:* ffiffiffiffiffi *εrr* p

*K*ð Þ*<sup>e</sup> <sup>c</sup>:r:* ffiffiffiffiffiffi *εrθ* p

And for *εr<sup>θ</sup>* <0, we obtain.

$$\begin{split} \varepsilon\_{r,\theta\overline{f}}^{TE} &= -|\ \varepsilon\_{r\theta}| \left( \mathbf{1} - \frac{\mathbf{1}}{|\ \varepsilon\_{r\theta}\ \mu\_{rx}| \mathbf{k}\_0^2} \cdot \left( \frac{u\_{nm}'}{R} \right)^2 \right) \\ &= -|\ \varepsilon\_{r\theta}| \left( \mathbf{1} - \left( \frac{f\_{c,nm}^{TE}}{f} \right)^2 \right) > \mathbf{0}, \text{ if } f < f\_{c,nm}^{TE}. \end{split} \tag{33}$$

Consequently, *μrz* <0 leads to *εTE <sup>r</sup>*,*eff* >0 below the cutoff frequency whenever *εr<sup>θ</sup>* >0 or *εr<sup>θ</sup>* <0.

Therefore, the relative permeability *μrz* below the cutoff frequency determines the sign of the relative effective permittivity of the anisotropic metamaterial in the circular waveguide. And the sign of the product *μrr:μrz* of the metamaterial below the cutoff frequency determines the sign of the propagation constants of the waveguide studied.

The backward waves are obtained for *μrr* <0 and *μrz* > 0 and the forward waves for *μrr* >0 and *μrz* < 0 and. Therefore, the backward waves and the forward waves can propagate below the cutoff frequency.

#### **2.2 Transverse magnetic (TM) modes**

Similar to TE modes, TM modes can be derived as follows: From Eq. (2), the differential equation for z-component can be obtained

$$\frac{\partial^2 E\_x}{\partial r^2} + \frac{\mathbf{1}}{r} \frac{\partial E\_x}{\partial r} + \left(\frac{K\_{c,r}^{(e)} \cdot \sqrt{\varepsilon\_{r\theta}}}{K\_{c,\theta}^{(e)} \cdot \sqrt{\varepsilon\_{rr}}}\right)^2 \frac{\mathbf{1}}{r^2} \frac{\partial^2 E\_x}{\partial \theta^2} + \left(\frac{\sqrt{\varepsilon\_{rz}}}{\sqrt{\varepsilon\_{rr}}} K\_{c,r}^{(e)}\right)^2 E\_x = \mathbf{0}. \tag{34}$$

Using the separation of the variables (*r,θ*), the expression of the longitudinal electric field *Ez* for the *TMnm* modes in the circular metallic waveguide completely filled with anisotropic metamaterial is necessary for the resolution of the differential Eq. (34). *Ez* can be written as follows

$$E\_x^{(\epsilon)} = E\_0 \cos\left(\frac{K\_{\epsilon,\theta}^{(\epsilon)} \sqrt{\varepsilon\_{rr}}}{K\_{\epsilon,r}^{(\epsilon)} \cdot \sqrt{\varepsilon\_{r\theta}}} n.\theta\right) I\_n\left(\frac{\sqrt{\varepsilon\_{rz}}}{\sqrt{\varepsilon\_{rr}}} K\_{\epsilon,r}^{(\epsilon)} .r\right) e^{-jk\_x x} \tag{35}$$

The expressions (5)–(8) become

$$E\_r^{(\epsilon)} = \frac{-jk\_\pm}{K\_{c.r}} \frac{\sqrt{\varepsilon\_{rz}}}{\sqrt{\varepsilon\_{rr}}} E\_0.\cos\left(\frac{K\_{c.\theta}^{(\epsilon)} \cdot \sqrt{\varepsilon\_{rr}}}{K\_{c.r}^{(\epsilon)} \cdot \sqrt{\varepsilon\_{r\theta}}} n.\theta\right) l\_n' \left(\frac{\sqrt{\varepsilon\_{rz}}}{\sqrt{\varepsilon\_{rr}}} K\_{c.r}^{(\epsilon)} r\right) e^{-jk\_\pm x} \tag{36}$$

$$E\_{\theta}^{(\epsilon)} = \frac{jk\_{\pi}}{K\_{c,\theta}K\_{c,r}} \frac{\sqrt{\varepsilon\_{rr}}}{\sqrt{\varepsilon\_{r\theta}}} \frac{n}{r} E\_{0}. \sin\left(\frac{K\_{c,\theta}^{(\epsilon)} \cdot \sqrt{\varepsilon\_{rr}}}{K\_{c,r}^{(\epsilon)} \cdot \sqrt{\varepsilon\_{r\theta}}} n. \theta\right) J\_{n}\left(\frac{\sqrt{\varepsilon\_{rz}}}{\sqrt{\varepsilon\_{rr}}} K\_{c,r}^{(\epsilon)} . r\right) e^{-jk\_{z}x} \tag{37}$$

*Rigorous Analysis of the Propagation in Metallic Circular Waveguide with Discontinuities… DOI: http://dx.doi.org/10.5772/intechopen.91645*

$$H\_r^{(\epsilon)} = \frac{-j\alpha\varepsilon\_0}{K\_{\epsilon,\theta}}\sqrt{\varepsilon\_{r\theta}}\sqrt{\varepsilon\_{rr}}\cdot\frac{n}{r}E\_0.\sin\left(\frac{K\_{\epsilon,\theta}^{(\epsilon)}\sqrt{\varepsilon\_{rr}}}{K\_{\epsilon,r}^{(\epsilon)}\sqrt{\varepsilon\_{r\theta}}}n.\theta\right)J\_n\left(\frac{\sqrt{\varepsilon\_{rz}}}{\sqrt{\varepsilon\_{rr}}}K\_{\epsilon,r}^{(\epsilon)}.r\right)e^{-jk\_\varepsilon x}\tag{38}$$

$$H\_{\theta}^{(\epsilon)} = \frac{-j\alpha\varepsilon\_{0}}{K\_{\varepsilon r}}\sqrt{\varepsilon\_{rr}}\sqrt{\varepsilon\_{rz}}E\_{0}.\cos\left(\frac{K\_{\varepsilon\theta}^{(\epsilon)}\sqrt{\varepsilon\_{rr}}}{K\_{\varepsilon r}^{(\epsilon)}\sqrt{\varepsilon\_{r\theta}}}n.\theta\right)l\_{n}^{\prime}\left(\frac{\sqrt{\varepsilon\_{rz}}}{\sqrt{\varepsilon\_{rr}}}K\_{\epsilon,r}^{(\epsilon)}.r\right)e^{-jk\_{z}x}\tag{39}$$

The boundary condition (18) gives the following equation

$$J\_n(\mathfrak{u}\_{nm}) = \mathbf{0}.\tag{40}$$

with

*2.1.2 Second case* μrz <0

*<sup>r</sup>*,*eff* is rewritten as

*<sup>r</sup>*,*eff* ¼ j j *εr<sup>θ</sup>* 1 þ

*<sup>r</sup>*,*eff* ¼ �j j *<sup>ε</sup>r<sup>θ</sup>* <sup>1</sup> � <sup>1</sup>

¼ �j j *<sup>ε</sup>r<sup>θ</sup>* <sup>1</sup> � *<sup>f</sup>*

@

Similar to TE modes, TM modes can be derived as follows:

*K*ð Þ*<sup>e</sup> <sup>c</sup>:<sup>θ</sup>:* ffiffiffiffiffi *εrr* p

!

*K*ð Þ*<sup>e</sup> <sup>c</sup>:<sup>r</sup>:* ffiffiffiffiffiffi *εrθ* p

> *K*ð Þ*<sup>e</sup> <sup>c</sup>:<sup>θ</sup>:* ffiffiffiffiffi *εrr* p

!

*K*ð Þ*<sup>e</sup> <sup>c</sup>:<sup>r</sup>:* ffiffiffiffiffiffi *εrθ* p

> *K*ð Þ*<sup>e</sup> <sup>c</sup>:<sup>θ</sup>:* ffiffiffiffiffi *εrr* p

*K*ð Þ*<sup>e</sup> <sup>c</sup>:<sup>r</sup>:* ffiffiffiffiffiffi *εrθ* p

*K*ð Þ*<sup>e</sup> <sup>c</sup>:<sup>r</sup>:* ffiffiffiffiffiffi *εrθ* p

*K*ð Þ*<sup>e</sup> <sup>c</sup>:<sup>θ</sup>:* ffiffiffiffiffi *εrr* p !<sup>2</sup>

From Eq. (2), the differential equation for z-component can be obtained

1 *r*2 *∂*2 *Ez ∂θ*<sup>2</sup> þ

Using the separation of the variables (*r,θ*), the expression of the longitudinal electric field *Ez* for the *TMnm* modes in the circular metallic waveguide completely filled with anisotropic metamaterial is necessary for the resolution of the differen-

*n:θ*

*n:θ*

!

*J* 0 *n*

*n:θ*

*Jn*

*Jn*

ffiffiffiffiffi *εrz* p ffiffiffiffiffi *εrr* <sup>p</sup> *<sup>K</sup>*ð Þ*<sup>e</sup> c:r* � �<sup>2</sup>

ffiffiffiffiffi *εrz* p ffiffiffiffiffi *εrr* <sup>p</sup> *<sup>K</sup>*ð Þ*<sup>e</sup> <sup>c</sup>:<sup>r</sup>:r* � �

> ffiffiffiffiffi *εrz* p ffiffiffiffiffi *εrr* <sup>p</sup> *<sup>K</sup>*ð Þ*<sup>e</sup> <sup>c</sup>:<sup>r</sup>:r* � �

> > ffiffiffiffiffi *εrz* p ffiffiffiffiffi *εrr* <sup>p</sup> *<sup>K</sup>*ð Þ*<sup>e</sup> <sup>c</sup>:<sup>r</sup>:r* � �

*e*

*e*

*e*

1 *<sup>ε</sup>r<sup>θ</sup> <sup>μ</sup>rz* j j*k*<sup>2</sup> 0 *: <sup>u</sup>*<sup>0</sup> *nm R*

*Electromagnetic Propagation and Waveguides in Photonics and Microwave Engineering*

*<sup>ε</sup>r<sup>θ</sup> <sup>μ</sup>rz* j j*k*<sup>2</sup> 0 *: <sup>u</sup>*<sup>0</sup> *nm R*

> *TE c:nm f*

Therefore, the relative permeability *μrz* below the cutoff frequency determines the sign of the relative effective permittivity of the anisotropic metamaterial in the circular waveguide. And the sign of the product *μrr:μrz* of the metamaterial below the cutoff frequency determines the sign of the propagation constants of the wave-

The backward waves are obtained for *μrr* <0 and *μrz* > 0 and the forward waves for *μrr* >0 and *μrz* < 0 and. Therefore, the backward waves and the forward waves

!<sup>2</sup> 0

� �2 !

1

A >0, if *f* < *f*

*<sup>r</sup>*,*eff* >0 below the cutoff frequency whenever

*TE*

� �2 !

>0*:* (32)

*<sup>c</sup>:nm:* (33)

*Ez* ¼ 0*:* (34)

�*jkzz* (35)

�*jkzz* (36)

�*jkzz* (37)

*εTE*

And for *εr<sup>θ</sup>* <0, we obtain.

*εTE*

Consequently, *μrz* <0 leads to *εTE*

can propagate below the cutoff frequency.

**2.2 Transverse magnetic (TM) modes**

1 *r ∂Ez ∂r* þ

tial Eq. (34). *Ez* can be written as follows

*<sup>z</sup>* ¼ *E*<sup>0</sup> cos

ffiffiffiffiffi *εrz* p ffiffiffiffiffi *εrr* p *E*0*:* cos

ffiffiffiffiffi *εrr* p ffiffiffiffiffiffi *εrθ* p

*n r E*0*:*sin

*E*ð Þ*<sup>e</sup>*

The expressions (5)–(8) become

For *εr<sup>θ</sup>* >0, *εTE*

*εr<sup>θ</sup>* >0 or *εr<sup>θ</sup>* <0.

guide studied.

*∂*2 *Ez ∂r*<sup>2</sup> þ

*E*ð Þ*<sup>e</sup> <sup>r</sup>* <sup>¼</sup> �*jkz Kc:<sup>r</sup>*

*<sup>θ</sup>* <sup>¼</sup> *jkz Kc:<sup>θ</sup>:Kc:<sup>r</sup>*

*E*ð Þ*<sup>e</sup>*

**70**

$$
\mu\_{nm} = \frac{\sqrt{\varepsilon\_{rz}}}{\sqrt{\varepsilon\_{rr}}} K\_{c.r.}^{(\epsilon)}.R. \tag{41}
$$

In Eq. (41) *unm* represents the mth zero (m = 1, 2, 3, … ) of the Bessel function *Jn* of the first kind of order n.

The constant *E*<sup>0</sup> is determined by normalizing the power flow down the circular guide.

$$P^{TM} = \int\_{0}^{R} \int\_{0} (E\_r^{(\epsilon)} H\_\theta^{\* (\epsilon)} - E\_\theta^{(\epsilon)} H\_r^{\* (\epsilon)}) r dr d\theta = 1 \tag{42}$$

Eq. (42) gives:

$$E\_0 = \frac{K\_{c.r}^2}{\sqrt{\alpha \varepsilon\_0 \varepsilon\_r k\_x}} N\_{nm}^{(e)} \tag{43}$$

with

$$N\_{nm}^{(\epsilon)} = \frac{1}{u\_{nm} J\_n'(u\_{nm}) \cdot \sqrt{\frac{\delta\_v}{2}}} \tag{44}$$

$$\delta\_n = \begin{cases} 2\pi, \text{ if } n = 0\\ \pi - \frac{\sin\left(4\pi b.\text{n}\right)}{4\text{b.n}}, \text{ if } n > 0 \end{cases} \tag{45}$$

$$b = \frac{K\_{c.\theta}^{(\epsilon)} \sqrt{\varepsilon\_{rr}}}{K\_{c.r.}^{(\epsilon)} \sqrt{\varepsilon\_{r\theta}}} \tag{46}$$

Finally, the propagation constant in TM mode is given by:

$$k\_{x.nm}^{(\text{TM})} = \pm \sqrt{k\_0^2 \varepsilon\_{rr} \mu\_{r\theta} - \frac{\varepsilon\_{rr}}{\varepsilon\_{rz}} \left(\frac{u\_{nm}}{R}\right)^2} \tag{47}$$

Obviously, the cutoff frequency is written

$$f\_{c.vm}^{(TM)} = \frac{c}{2\pi} \frac{\mathbf{1}}{\sqrt{|\mu\_{r0}\varepsilon\_{rz}|}} \cdot \left(\frac{\mu\_{nm}}{R}\right). \tag{48}$$

We can introduce the following effective permeability and effective permittivity to describe the propagation characteristics of the waveguide modes.

*Electromagnetic Propagation and Waveguides in Photonics and Microwave Engineering*

$$
\varepsilon\_{r, \sharp \overline{f}}^{\text{TM}} = \varepsilon\_{rr}, \tag{49}
$$

It is also seen that the relative permittivity *εrz* which is independent of *μr<sup>θ</sup>*

*Rigorous Analysis of the Propagation in Metallic Circular Waveguide with Discontinuities…*

sign of *εrz* and *εrr* for TM modes and the sign of *μrz* and *μrr* for TE modes.

**2.3 Analysis of uniaxial discontinuities in the circular waveguides**

tinuity in waveguide. The discontinuities are considered without losses.

*ET* <sup>¼</sup> <sup>X</sup><sup>∞</sup> *m*¼1 *Ai <sup>m</sup> ai <sup>m</sup>* <sup>þ</sup> *<sup>b</sup><sup>i</sup> m* � �*e*

*HT* <sup>¼</sup> <sup>X</sup><sup>∞</sup>

modal Eigen function in the guide *i*, respectively and *Ai*

guides (m is the index of the mode and *i = I, II*).

*m*¼1 *Bi <sup>m</sup> a<sup>i</sup> <sup>m</sup>* � *bi m* � �*hi*

same cross section filled with two different media. *ai* and *b<sup>i</sup>*

reflected waves, respectively.

written in the modal bases as follows [20]:

components in the transverse plane), *h<sup>i</sup>*

tions:

**Figure 2.**

**73**

metamaterial in the circular waveguide. The forward wave propagates in the waveguide for *εrz* < 0 and *εrr* > 0, and backward wave propagates for *εrz* >0 and *εrr* <0. Therefore from this analysis, it is found that both the backward waves and the forward waves can propagate in any frequency region. This is determined by the

In this section, we analyzed a waveguide filters filled with partially anisotropic metamaterial using the extension of the mode matching technique based on the Scattering Matrix Approach which, from the decomposition of the modal fields, are used to determine the dispersion matrix and thus the characterization of a discon-

In **Figure 2** we consider a junction between two circular waveguides having the

The transverse electric and magnetic fields (*ET*, *HT*) in the wave guides can be

where *HT* and *ET* are the transverse magnetic and electric fields (*T* refers to the

*<sup>m</sup>*, *e<sup>i</sup>*

coefficients which are determined by normalizing the power flow down the circular

At the junction, the continuity of the fields allows to write the following equa-

*EI <sup>t</sup>* <sup>¼</sup> *EII*

*HI <sup>t</sup>* <sup>¼</sup> *<sup>H</sup>II*

*Junction between two circular waveguides filled with two different media having the same cross section.*

*i*

*<sup>r</sup>*,*eff* of the anisotropic

are the incident and the

*<sup>m</sup>* (55)

*<sup>m</sup>* (56)

*<sup>m</sup>* are complex

*<sup>m</sup>* represent the mth magnetic and electric

*<sup>t</sup>* (57)

*<sup>t</sup>* (58)

*<sup>m</sup>* and *Bi*

determines the sign of the relative effective permeability *μTM*

*DOI: http://dx.doi.org/10.5772/intechopen.91645*

$$
\mu\_{r, \text{eff}}^{\text{TM}} = \mu\_{r\theta} \left( 1 - \frac{\mathbf{1}}{\mu\_{r\theta} \varepsilon\_{rx} k\_0^2} \cdot \left( \frac{u\_{nm}}{R} \right)^2 \right). \tag{50}
$$

Similar to the previous discussion, we have three possibilities: Further, It is apparent that:

$$\begin{aligned} \bullet \; k\_x^{TM} &= k\_0 \sqrt{\mu\_{r, \text{eff}}^{TM} \cdot \varepsilon\_{r, \text{eff}}^{TM}} > 0, \text{ for } \mu\_{r, \text{eff}}^{TM} > 0 \text{ and } \varepsilon\_{r, \text{eff}}^{TM} > 0; \\\\ \bullet \; k\_x^{TM} &= -k\_0 \sqrt{\mu\_{r, \text{eff}}^{TM} \cdot \varepsilon\_{r, \text{eff}}^{TM}} < 0, \text{ for } \mu\_{r, \text{eff}}^{TM} < 0 \text{ and } \varepsilon\_{r, \text{eff}}^{TM} < 0; \\\\ \bullet \; k\_x^{TM} &= -k\_0 \sqrt{\mu\_{r, \text{eff}}^{TM} \cdot \varepsilon\_{r, \text{eff}}^{TM}} < \sqrt{\mu\_{r, \text{eff}}^{TM} \cdot \varepsilon\_{r, \text{eff}}^{TM}} < 0. \end{aligned}$$

$$\bullet \ k\_x^{TM} = \pm jk\_0 \sqrt{\mu\_{r, \epsilon \emptyset \prime}^{TM} \varepsilon\_{r, \epsilon \emptyset \prime}^{TM}}, \text{ for } \mu\_{r, \epsilon \emptyset \prime}^{TM} . \varepsilon\_{r, \epsilon \emptyset \prime}^{TM} < 0.$$

Consequently, the sign of *μTM <sup>r</sup>*,*eff* depends on the sign of *εrz*. In the following, we will consider all cases that arise from the different sign of *εrz*.

## *2.2.1 Case when* εrz >0

In this case, for *μr<sup>θ</sup>* >0, *μTM <sup>r</sup>*,*eff* is rewritten as.

$$|\mu\_{r,\ell\overline{\mathcal{J}}}^{\rm TM} = |\mu\_{r\theta}| \left( 1 - \frac{1}{|\mu\_{r\theta} \,\varepsilon\_{rz}| \overline{k\_0}^2}, \left( \frac{\mu\_{nm}}{R} \right)^2 \right) = |\mu\_{r\theta}| \left( 1 - \left( \frac{f\_{c.nm}^{\rm TM}}{f} \right)^2 \right) < 0, \text{ if } f < f\_{c.nm}^{\rm TM} \tag{51}$$

And for *μr<sup>θ</sup>* < 0, we have

$$\mu\_{r,\xi\overline{f}}^{\text{TM}} = -|\,\mu\_{r\theta}| \left( \mathbf{1} + \frac{\mathbf{1}}{|\,\mu\_{r\theta}\varepsilon\_{r\overline{z}}|k\_0^2} \cdot \left(\frac{u\_{nm}}{R}\right)^2 \right) < \mathbf{0}.\tag{52}$$

It can be seen that *εrz* > 0 leads to *μTM <sup>r</sup>*,*eff* < 0 below the cutoff frequency whenever *μr<sup>θ</sup>* >0, or *μr<sup>θ</sup>* <0.

### *2.2.2 Case when* εrz <0

In this case, for *μr<sup>θ</sup>* >0, we have

$$|\mu\_{r,\emptyset}^{TM}| = |\mu\_{r\theta}| \left( 1 + \frac{1}{|\mu\_{r\theta}| \varepsilon\_{rz} |k\_0^2|} \cdot \left( \frac{u\_{nm}}{R} \right)^2 \right) > 0. \tag{53}$$

and for *μr<sup>θ</sup>* <0, we obtain.

$$\mu\_{r\mathcal{eff}}^{\rm TM} = -|\,\mu\_{r\theta}| \left( \mathbf{1} - \frac{\mathbf{1}}{|\,\mu\_{r\theta}\,\varepsilon\_{r\mathbf{z}}| k\_0^2}, \left(\frac{\mu\_{nm}}{R}\right)^2 \right) = -|\,\mu\_{r\theta}| \left( \mathbf{1} - \left(\frac{f\_{c.m\mathbf{n}}^{\rm TM}}{f}\right)^2 \right) > 0\,,\ \text{if}\, f \prec f\_{c.m\mathbf{n}}^{\rm TM} \tag{54}$$

*Rigorous Analysis of the Propagation in Metallic Circular Waveguide with Discontinuities… DOI: http://dx.doi.org/10.5772/intechopen.91645*

It is also seen that the relative permittivity *εrz* which is independent of *μr<sup>θ</sup>* determines the sign of the relative effective permeability *μTM <sup>r</sup>*,*eff* of the anisotropic metamaterial in the circular waveguide. The forward wave propagates in the waveguide for *εrz* < 0 and *εrr* > 0, and backward wave propagates for *εrz* >0 and *εrr* <0.

Therefore from this analysis, it is found that both the backward waves and the forward waves can propagate in any frequency region. This is determined by the sign of *εrz* and *εrr* for TM modes and the sign of *μrz* and *μrr* for TE modes.

#### **2.3 Analysis of uniaxial discontinuities in the circular waveguides**

In this section, we analyzed a waveguide filters filled with partially anisotropic metamaterial using the extension of the mode matching technique based on the Scattering Matrix Approach which, from the decomposition of the modal fields, are used to determine the dispersion matrix and thus the characterization of a discontinuity in waveguide. The discontinuities are considered without losses.

In **Figure 2** we consider a junction between two circular waveguides having the same cross section filled with two different media. *ai* and *b<sup>i</sup>* are the incident and the reflected waves, respectively.

The transverse electric and magnetic fields (*ET*, *HT*) in the wave guides can be written in the modal bases as follows [20]:

$$E\_T = \sum\_{m=1}^{\infty} A\_m^i \left( a\_m^i + b\_m^i \right) e\_m^i \tag{55}$$

$$H\_T = \sum\_{m=1}^{\infty} B\_m^i \left(a\_m^i - b\_m^i\right) h\_m^i \tag{56}$$

where *HT* and *ET* are the transverse magnetic and electric fields (*T* refers to the components in the transverse plane), *h<sup>i</sup> <sup>m</sup>*, *e<sup>i</sup> <sup>m</sup>* represent the mth magnetic and electric modal Eigen function in the guide *i*, respectively and *Ai <sup>m</sup>* and *Bi <sup>m</sup>* are complex coefficients which are determined by normalizing the power flow down the circular guides (m is the index of the mode and *i = I, II*).

At the junction, the continuity of the fields allows to write the following equations:

$$E\_t^l = E\_t^{ll} \tag{57}$$

$$H\_t^I = H\_t^{\text{II}} \tag{58}$$

**Figure 2.** *Junction between two circular waveguides filled with two different media having the same cross section.*

*εTM*

*Electromagnetic Propagation and Waveguides in Photonics and Microwave Engineering*

*μrθεrzk*<sup>2</sup> 0

*<sup>r</sup>*,*eff* >0 and *εTM*

*<sup>r</sup>*,*eff* <0*:*

*<sup>r</sup>*,*eff* <0 and *εTM*

!

*: unm R* � �<sup>2</sup>

*<sup>r</sup>*,*eff* >0;

<sup>¼</sup> *<sup>μ</sup>r<sup>θ</sup>* j j <sup>1</sup> � *<sup>f</sup>*

@

*: unm R* � �<sup>2</sup>

*: unm R* � �<sup>2</sup>

¼ � *<sup>μ</sup>r<sup>θ</sup>* j j <sup>1</sup> � *<sup>f</sup>*

@

1 *<sup>μ</sup>r<sup>θ</sup>* j j *<sup>ε</sup>rz <sup>k</sup>*<sup>2</sup> 0

1 *<sup>μ</sup>r<sup>θ</sup>* j j *<sup>ε</sup>rz <sup>k</sup>*<sup>2</sup> 0

!

!

*<sup>r</sup>*,*eff* <0;

*<sup>r</sup>*,*eff* depends on the sign of *εrz*. In the following, we

*TM c:nm f*

*<sup>r</sup>*,*eff* < 0 below the cutoff frequency whenever

*TM c:nm f*

1

!<sup>2</sup> 0

1

A <0, if *f* < *f*

<0*:* (52)

>0*:* (53)

A >0*:*, if *f* < *f*

*TM c:nm*

(54)

*TM c:nm*

(51)

!<sup>2</sup> 0

*<sup>r</sup>*,*eff* <sup>¼</sup> *<sup>μ</sup>r<sup>θ</sup>* <sup>1</sup> � <sup>1</sup>

Similar to the previous discussion, we have three possibilities:

> 0, for *μTM*

<0, for *μTM*

, for *μTM*

will consider all cases that arise from the different sign of *εrz*.

*: unm R* � �<sup>2</sup>

*<sup>r</sup>*,*eff* ¼ � *μr<sup>θ</sup>* j j 1 þ

*<sup>r</sup>*,*eff* ¼ *μr<sup>θ</sup>* j j 1 þ

*: unm R* � �<sup>2</sup>

*<sup>r</sup>*,*eff :εTM*

*<sup>r</sup>*,*eff* is rewritten as.

*μTM*

Further, It is apparent that:

q

q

Consequently, the sign of *μTM*

In this case, for *μr<sup>θ</sup>* >0, *μTM*

*<sup>r</sup>*,*eff* <sup>¼</sup> *<sup>μ</sup>r<sup>θ</sup>* j j <sup>1</sup> � <sup>1</sup>

And for *μr<sup>θ</sup>* < 0, we have

*μr<sup>θ</sup>* >0, or *μr<sup>θ</sup>* <0.

*μTM*

**72**

*2.2.2 Case when* εrz <0

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *μTM <sup>r</sup>*,*eff :εTM r*,*eff*

> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *μTM <sup>r</sup>*,*eff :εTM r*,*eff*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *μTM <sup>r</sup>*,*eff :εTM r*,*eff*

> *<sup>μ</sup>r<sup>θ</sup>* j j *<sup>ε</sup>rz <sup>k</sup>*<sup>2</sup> 0

*μTM*

It can be seen that *εrz* > 0 leads to *μTM*

In this case, for *μr<sup>θ</sup>* >0, we have

and for *μr<sup>θ</sup>* <0, we obtain.

*<sup>r</sup>*,*eff* ¼ � *<sup>μ</sup>r<sup>θ</sup>* j j <sup>1</sup> � <sup>1</sup>

*μTM*

*<sup>μ</sup>r<sup>θ</sup>* j j *<sup>ε</sup>rz <sup>k</sup>*<sup>2</sup> 0

!

!

• *kTM <sup>z</sup>* ¼ *k*<sup>0</sup>

• *kTM*

• *kTM*

*μTM*

*<sup>z</sup>* ¼ �*k*<sup>0</sup>

*<sup>z</sup>* ¼ �*jk*<sup>0</sup>

*2.2.1 Case when* εrz >0

*<sup>r</sup>*,*eff* ¼ *εrr*, (49)

*:* (50)

By postponing the Eqs. (55) and (56) in (57) and (58), we obtain:

$$\sum\_{m=1}^{N\_1} A\_m^I (a\_m^I + b\_m^I) e\_m^I = \sum\_{p=1}^{N\_2} A\_p^{II} \left(a\_p^{II} + b\_p^{II}\right) e\_p^{II} \tag{59}$$

The scattering matrix of the discontinuity is:

*DOI: http://dx.doi.org/10.5772/intechopen.91645*

**3. Numerical results and discussion**

**3.1 Propagating modes**

*<sup>c</sup>:*<sup>11</sup> ¼ 3*:*13 *GHz*.

of *f TE*

waves).

**Figure 3.**

**75**

*Curves of propagation constant* kTE

*metamaterial with parameters* μrr ¼ *1,* μrz ¼ �*1,* ε<sup>r</sup><sup>θ</sup> ¼ *4:4.*

*<sup>S</sup>* <sup>¼</sup> *U M*<sup>1</sup> �*M*<sup>2</sup> *U*

We choose the radius of the circular metal guide R = 13.4 mm.

�<sup>1</sup> *U M*<sup>1</sup>

*Rigorous Analysis of the Propagation in Metallic Circular Waveguide with Discontinuities…*

The total scattering matrix is obtained by chaining the S scattering matrices of all the discontinuities in a waveguide having cascaded uniaxial discontinuities [21].

In a first case, we study the TE modes of a circular guide completely filled with

It is interesting to see that both forward and backward waves can be obtained by controlling the signs of *μrz* and *μrr*. Our results agree well with the predicted ones. In a second case, we study the TE modes of this circular waveguide. **Figure 5** represents the curves of propagation constant for the frequency range 1–10 GHz

<sup>z</sup> *for TE mode of the circular waveguide completely filled anisotropic*

anisotropic metamaterials (see **Figure 1**) with negative *μrr* or negative *μrz*. The fundamental mode of the equivalent empty circular waveguide has a resonant frequency of 6.57 GHz. For the case of metamaterials with a permeability *μ<sup>r</sup>* ¼ �1 and permittivity *ε<sup>r</sup>* ¼ �4*:*4, the fundamental mode presents a resonance frequency

In **Figure 3** the curves of the propagation constant, for frequency range 1–10 GHz and for the first five TE modes with *μrr* ¼ 1, *μrz* ¼ �1 and *εr<sup>θ</sup>* ¼ 4*:*4, are represented. We observe that all modes propagate without cutoff frequencies (forward waves). **Figure 4** represents the same diagrams for *μrr* ¼ �1, *μrz* ¼ 1 and *εr<sup>θ</sup>* ¼ 4*:*4. When *n* and *m* are small and *ω* is large, the waves stop propagating. So, these modes propagate at low frequencies and cutoff at high frequencies (backward

*M*<sup>2</sup> �*U*

(69)

$$\sum\_{m=1}^{N\_1} B\_m^I \left( a\_m^I - b\_m^I \right) h\_m^I = \sum\_{p=1}^{N\_2} B\_p^{II} \left( -a\_p^{II} + b\_p^{II} \right) h\_p^{II} \tag{60}$$

N1 and N2 are the number of considered modes in guides 1 and 2, respectively. By applying the Galerkin method, Eqs. (59) and (60), lead to the following systems:

$$\sum\_{m=1}^{N\_1} A\_m^I \left( a\_m^I + b\_m^I \right) \left< e\_m^I | e\_p^{II} \right> = A\_p^{II} \left( a\_p^{II} + b\_p^{II} \right) \tag{61}$$

$$B\_m^I \left( a\_m^I - b\_m^I \right) = \sum\_{p=1}^{N\_2} B\_p^{II} \left( -a\_p^{II} + b\_p^{II} \right) \left\langle h\_p^{II} | h\_m^I \right\rangle \tag{62}$$

The inner product is defined as:

$$
\langle e\_m | e\_p \rangle = \int\_S e\_m^\* e\_p \, d\mathbf{S} \tag{63}
$$

The Eqs. (61) and (62) give:

$$-a\_p^{\ II} + \sum\_{m=1}^{N\_1} \frac{A\_m^{I}}{A\_p^{II}} a\_m^I \left< e\_m^I | e\_p^{II} \right> = b\_p^{\ II} - \sum\_{m=1}^{N\_1} \frac{A\_m^{I}}{A\_p^{II}} b\_m^I \left< e\_m^I | e\_p^{II} \right> \tag{64}$$

$$a\_m^I + \sum\_{p=1}^{N\_2} \frac{B\_p^{II}}{B\_m^I} a\_p^{II} \left< h\_p^{II} | h\_m^I \right> = b\_m^I + \sum\_{p=1}^{N\_2} \frac{B\_p^{II}}{B\_m^I} b\_p^{II} \left< h\_p^{II} | h\_m^I \right> \tag{65}$$

which can be written in matrix form:

$$
\begin{bmatrix} U & M\_1 \\ & \begin{bmatrix} U & & \\ & & \\ & & & \end{bmatrix} \\ & & & & \begin{bmatrix} & & & & \\ & & & & & & \\ & & & & & & \end{bmatrix} \end{bmatrix} = \begin{bmatrix} U & M\_1 \\ & \begin{bmatrix} U & & & & & \\ & \begin{bmatrix} & & & & \\ & & & & & \end{bmatrix} \\ & & & & & \begin{bmatrix} & & & & \\ & & & & & \end{bmatrix} \end{bmatrix} \tag{66}
$$

where *U* is the identity matrix. *M1* and *M2* are defined as:

$$\mathbf{M}\_{\mathbf{1}\dot{\boldsymbol{y}}} = \frac{\mathbf{B}\_{\dot{\boldsymbol{y}}}^{\mathrm{II}}}{\mathbf{B}\_{\dot{\boldsymbol{y}}}^{\mathrm{I}}} \left< h\_{\boldsymbol{y}}^{\mathrm{II}} \middle| h\_{\boldsymbol{i}}^{\mathrm{I}} \right> \tag{67}$$

$$\mathbf{M}\_{2\ddagger} = \frac{\mathbf{A}\_i^I}{\mathbf{A}\_j^{II}} \left\langle e\_i^I \middle| e\_j^{II} \right\rangle \tag{68}$$

*Rigorous Analysis of the Propagation in Metallic Circular Waveguide with Discontinuities… DOI: http://dx.doi.org/10.5772/intechopen.91645*

The scattering matrix of the discontinuity is:

$$\mathbf{S} = \begin{bmatrix} U & M\_1 \\ -M\_2 & U \end{bmatrix}^{-1} \begin{bmatrix} U & M\_1 \\ M\_2 & -U \end{bmatrix} \tag{69}$$

The total scattering matrix is obtained by chaining the S scattering matrices of all the discontinuities in a waveguide having cascaded uniaxial discontinuities [21].

### **3. Numerical results and discussion**

#### **3.1 Propagating modes**

By postponing the Eqs. (55) and (56) in (57) and (58), we obtain:

*Electromagnetic Propagation and Waveguides in Photonics and Microwave Engineering*

*I <sup>m</sup>* <sup>¼</sup> <sup>X</sup> *N*<sup>2</sup>

*p*¼1

*<sup>m</sup>* <sup>¼</sup> <sup>X</sup> *N*<sup>2</sup>

*p*¼1 *BII <sup>p</sup>* �*aII*

N1 and N2 are the number of considered modes in guides 1 and 2, respectively. By applying the Galerkin method, Eqs. (59) and (60), lead to the following systems:

> *I m*j*e II p* D E

> > *S e* ∗

<sup>¼</sup> *<sup>b</sup>II*

<sup>¼</sup> *bI*

*<sup>p</sup>* �<sup>X</sup> *N*<sup>1</sup>

*<sup>m</sup>* <sup>þ</sup><sup>X</sup> *N*<sup>2</sup>

<sup>¼</sup> *U M*<sup>1</sup> �*M*<sup>2</sup> *U* � �

> *hII j* D

*hI i* �

*m*¼1

*p*¼1

*AI m AII p bI <sup>m</sup> e I m*j*e II p* D E

*BII p BI m bII <sup>p</sup> hII <sup>p</sup>* <sup>j</sup>*hI m* D E

> *bI* 1 *: bI N*<sup>1</sup> *bII* 1 *: bII N*<sup>2</sup>

� � (67)

*N*<sup>2</sup>

*p*¼1 *BII <sup>p</sup>* �*aII*

h*em ep* � � � ¼ ð

> *aI* 1 *: aI N*<sup>1</sup> *aII* 1 *: aII N*<sup>2</sup>

where *U* is the identity matrix. *M1* and *M2* are defined as:

*<sup>M</sup>*1*ij* <sup>¼</sup> *BII j BI i*

*<sup>M</sup>*2*ij* <sup>¼</sup> *<sup>A</sup><sup>I</sup> i AII j e I i* � *e II j* � � � E

*AII <sup>p</sup> aII <sup>p</sup>* <sup>þ</sup> *bII p* � �

<sup>¼</sup> *AII <sup>p</sup> aII <sup>p</sup>* <sup>þ</sup> *<sup>b</sup>II p* � �

*<sup>p</sup>* <sup>þ</sup> *<sup>b</sup>II p* � � *e II*

*hII*

*<sup>p</sup>* <sup>þ</sup> *<sup>b</sup>II p* � �

> *hII <sup>p</sup>* <sup>j</sup>*h<sup>I</sup> m* D E

*mep dS* (63)

*<sup>p</sup>* (59)

*<sup>p</sup>* (60)

(61)

(62)

(64)

(65)

(66)

(68)

X *N*<sup>1</sup>

*m*¼1 *AI <sup>m</sup> aI <sup>m</sup>* <sup>þ</sup> *<sup>b</sup><sup>I</sup> m* � �*e*

X *N*<sup>1</sup>

*m*¼1 *BI <sup>m</sup> a<sup>I</sup> <sup>m</sup>* � *bI m* � �*h<sup>I</sup>*

X *N*<sup>1</sup>

*m*¼1 *AI <sup>m</sup> aI <sup>m</sup>* <sup>þ</sup> *<sup>b</sup><sup>I</sup> m* � � *e*

*BI <sup>m</sup> a<sup>I</sup> <sup>m</sup>* � *bI m* � � <sup>¼</sup> <sup>X</sup>

The inner product is defined as:

The Eqs. (61) and (62) give:

�*aII*

*aI <sup>m</sup>* <sup>þ</sup><sup>X</sup> *N*<sup>2</sup>

**74**

*<sup>p</sup>* <sup>þ</sup><sup>X</sup> *N*<sup>1</sup>

*m*¼1

*p*¼1

which can be written in matrix form:

*AI m AII p aI <sup>m</sup> e I m*j*e II p* D E

*BII p BI m aII <sup>p</sup> <sup>h</sup>II <sup>p</sup>* <sup>j</sup>*h<sup>I</sup> m* D E

*U M*<sup>1</sup> *M*<sup>2</sup> �*U* � � We choose the radius of the circular metal guide R = 13.4 mm.

In a first case, we study the TE modes of a circular guide completely filled with anisotropic metamaterials (see **Figure 1**) with negative *μrr* or negative *μrz*. The fundamental mode of the equivalent empty circular waveguide has a resonant frequency of 6.57 GHz. For the case of metamaterials with a permeability *μ<sup>r</sup>* ¼ �1 and permittivity *ε<sup>r</sup>* ¼ �4*:*4, the fundamental mode presents a resonance frequency of *f TE <sup>c</sup>:*<sup>11</sup> ¼ 3*:*13 *GHz*.

In **Figure 3** the curves of the propagation constant, for frequency range 1–10 GHz and for the first five TE modes with *μrr* ¼ 1, *μrz* ¼ �1 and *εr<sup>θ</sup>* ¼ 4*:*4, are represented. We observe that all modes propagate without cutoff frequencies (forward waves). **Figure 4** represents the same diagrams for *μrr* ¼ �1, *μrz* ¼ 1 and *εr<sup>θ</sup>* ¼ 4*:*4. When *n* and *m* are small and *ω* is large, the waves stop propagating. So, these modes propagate at low frequencies and cutoff at high frequencies (backward waves).

It is interesting to see that both forward and backward waves can be obtained by controlling the signs of *μrz* and *μrr*. Our results agree well with the predicted ones.

In a second case, we study the TE modes of this circular waveguide. **Figure 5** represents the curves of propagation constant for the frequency range 1–10 GHz

#### **Figure 3.**

*Curves of propagation constant* kTE <sup>z</sup> *for TE mode of the circular waveguide completely filled anisotropic metamaterial with parameters* μrr ¼ *1,* μrz ¼ �*1,* ε<sup>r</sup><sup>θ</sup> ¼ *4:4.*

#### **Figure 4.**

*Curves of propagation constant* kTE <sup>z</sup> *for TE mode of the circular waveguide completely filled anisotropic metamaterial with parameters* μrr ¼ �*1,* μrz ¼ *1 and* ε<sup>r</sup><sup>θ</sup> ¼ *4:4.*

and for the first five TM modes with *εrr* ¼ 4*:*4, *εrz* ¼ �4*:*4 and *μr<sup>θ</sup>* ¼ 1. All modes propagate without cutoff (forward waves).

Calculated curves of propagation constant for the frequency range 1–10 GHz and for the first five TM modes with *εrr* ¼ �4*:*4, *εrz* ¼ 4*:*4, *μr<sup>θ</sup>* ¼ 1 are presented. We notice that both forward wave and backward wave can be obtained by controlling the signs of *εrr* and *εrz*. **Figures 5** and **6** show that our results agree well with the predicted ones.

We observe that the cutoff frequencies of lowest TE modes decreased with the respect increase of *μrz* for *μrr* ¼ �1 and *εr<sup>θ</sup>* ¼ 4*:*4 (see **Figure 7**). In a same manner, the TM cutoff frequencies decreased with the respect increase of *εrz* for *εrr* ¼ �4*:*4 and *μr<sup>θ</sup>* ¼ 1 (see **Figure 8**). Consequently, by varying the parameters of material the propagating mode can be controlled.

**3.2 Filter design**

**Figure 7.**

**Figure 6.**

*Curves of propagation constant* kTM

*metamaterial with parameters* εrr ¼ �*4:4,* εrz ¼ *4:4,* μ<sup>r</sup><sup>θ</sup> ¼ *1.*

*DOI: http://dx.doi.org/10.5772/intechopen.91645*

studied structure.

**77**

We consider now, 12 discontinuities (see **Figure 9**) constituted by juxtaposing 13 circular waveguides having the same dimensions (R = 13.4 mm). The circuit is formed by alternation of empty guide (*ε<sup>r</sup>* ¼ *μ<sup>r</sup>* ¼ 1) of width l = 10 mm and guide filled by anisotropic metamaterials (*εrr* ¼ *εr<sup>θ</sup>* ¼ �*εrz* ¼ �4*:*4; *μrr* ¼ *μr<sup>θ</sup>* ¼ �*μrz* ¼ 1) of width d = 0.2 mm (periodic structure). **Figure 9** represents the geometry of the

*The cutoff frequencies for the first five TE modes versus* μrz *with* μrr ¼ �*1,* ε<sup>r</sup><sup>θ</sup> ¼ *4:4.*

<sup>z</sup> *for TM mode of the circular waveguide completely filled anisotropic*

*Rigorous Analysis of the Propagation in Metallic Circular Waveguide with Discontinuities…*

The transmission and reflection coefficients using our numerical method with MATLAB and HFSS are presented in **Figure 10**. We used 8 modes in the whole circuit for the modal method. The simulations results show that are in perfect agreement. However and especially if the number of discontinuities increases, our method is significantly faster than HFSS. Then, by using our approach, it could easy

to design filters according to a given specifications.

#### **Figure 5.**

*Curves of propagation constant* kTM <sup>z</sup> *for TM mode of the circular waveguide completely filled anisotropic metamaterial with parameters* εrr ¼ *4:4,* εrz ¼ �*4:4,* μ<sup>r</sup><sup>θ</sup> ¼ *1.*

*Rigorous Analysis of the Propagation in Metallic Circular Waveguide with Discontinuities… DOI: http://dx.doi.org/10.5772/intechopen.91645*

#### **Figure 6.**

and for the first five TM modes with *εrr* ¼ 4*:*4, *εrz* ¼ �4*:*4 and *μr<sup>θ</sup>* ¼ 1. All modes

*Electromagnetic Propagation and Waveguides in Photonics and Microwave Engineering*

Calculated curves of propagation constant for the frequency range 1–10 GHz and for the first five TM modes with *εrr* ¼ �4*:*4, *εrz* ¼ 4*:*4, *μr<sup>θ</sup>* ¼ 1 are presented. We notice that both forward wave and backward wave can be obtained by controlling the signs of *εrr* and *εrz*. **Figures 5** and **6** show that our results agree well with the

<sup>z</sup> *for TE mode of the circular waveguide completely filled anisotropic*

<sup>z</sup> *for TM mode of the circular waveguide completely filled anisotropic*

We observe that the cutoff frequencies of lowest TE modes decreased with the respect increase of *μrz* for *μrr* ¼ �1 and *εr<sup>θ</sup>* ¼ 4*:*4 (see **Figure 7**). In a same manner, the TM cutoff frequencies decreased with the respect increase of *εrz* for *εrr* ¼ �4*:*4 and *μr<sup>θ</sup>* ¼ 1 (see **Figure 8**). Consequently, by varying the parameters of material the

propagate without cutoff (forward waves).

*metamaterial with parameters* μrr ¼ �*1,* μrz ¼ *1 and* ε<sup>r</sup><sup>θ</sup> ¼ *4:4.*

propagating mode can be controlled.

predicted ones.

**Figure 5.**

**76**

*Curves of propagation constant* kTM

*metamaterial with parameters* εrr ¼ *4:4,* εrz ¼ �*4:4,* μ<sup>r</sup><sup>θ</sup> ¼ *1.*

**Figure 4.**

*Curves of propagation constant* kTE

*Curves of propagation constant* kTM <sup>z</sup> *for TM mode of the circular waveguide completely filled anisotropic metamaterial with parameters* εrr ¼ �*4:4,* εrz ¼ *4:4,* μ<sup>r</sup><sup>θ</sup> ¼ *1.*

**Figure 7.** *The cutoff frequencies for the first five TE modes versus* μrz *with* μrr ¼ �*1,* ε<sup>r</sup><sup>θ</sup> ¼ *4:4.*

#### **3.2 Filter design**

We consider now, 12 discontinuities (see **Figure 9**) constituted by juxtaposing 13 circular waveguides having the same dimensions (R = 13.4 mm). The circuit is formed by alternation of empty guide (*ε<sup>r</sup>* ¼ *μ<sup>r</sup>* ¼ 1) of width l = 10 mm and guide filled by anisotropic metamaterials (*εrr* ¼ *εr<sup>θ</sup>* ¼ �*εrz* ¼ �4*:*4; *μrr* ¼ *μr<sup>θ</sup>* ¼ �*μrz* ¼ 1) of width d = 0.2 mm (periodic structure). **Figure 9** represents the geometry of the studied structure.

The transmission and reflection coefficients using our numerical method with MATLAB and HFSS are presented in **Figure 10**. We used 8 modes in the whole circuit for the modal method. The simulations results show that are in perfect agreement. However and especially if the number of discontinuities increases, our method is significantly faster than HFSS. Then, by using our approach, it could easy to design filters according to a given specifications.

propagation constant of the waveguide are closely dependent on constitutive parameters of the metamaterial. Using our MATLAB code the dispersion curves of the fundamental mode and the first four higher order modes of the metamaterial

*Rigorous Analysis of the Propagation in Metallic Circular Waveguide with Discontinuities…*

We found that in different frequency ranges below and above the cutoff frequency both the forward and the backward waves can propagate. This is determined by the sign of *εrz* and *εrr* for TM modes and by the sign of *μrz* and *μrr* for TE modes. Our simulation results are in good agreement with the theoretical

Moreover, using the Scattering Matrix Approach we applied the extension of MM technique to determine the dispersion matrix and to analyze multiple uniaxial circular discontinuity in waveguide filled with anisotropic metamaterials. This introduced tool is applied to the modeling of large complex structures such as filters

where its rapidity compared to the commercial simulation tools is verified.

1 MACS Research Laboratory, National Engineering School of Gabes, Gabes

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

waveguide are obtained.

*DOI: http://dx.doi.org/10.5772/intechopen.91645*

prediction.

**Author details**

**79**

Hedi Sakli1,2\* and Wyssem Fathallah1

2 EITA Consulting, Montesson France

provided the original work is properly cited.

\*Address all correspondence to: hedi.s@eitaconsulting.fr

University, Gabes, Tunisia

**Figure 8.** *The cutoff frequencies for the first five TM modes versus* εrz *with* εrr ¼ �*4:4,* μ<sup>r</sup><sup>θ</sup> ¼ *1.*

**Figure 9.** *Geometry of the circular waveguide with 12 discontinuities.*

#### **Figure 10.**

*Reflection coefficient of the periodic structure with 12 discontinuities.*

### **4. Conclusion**

Rigorous analysis of propagating modes in circular waveguides filled with anisotropic metamaterial has been developed. It was demonstrated that the

*Rigorous Analysis of the Propagation in Metallic Circular Waveguide with Discontinuities… DOI: http://dx.doi.org/10.5772/intechopen.91645*

propagation constant of the waveguide are closely dependent on constitutive parameters of the metamaterial. Using our MATLAB code the dispersion curves of the fundamental mode and the first four higher order modes of the metamaterial waveguide are obtained.

We found that in different frequency ranges below and above the cutoff frequency both the forward and the backward waves can propagate. This is determined by the sign of *εrz* and *εrr* for TM modes and by the sign of *μrz* and *μrr* for TE modes. Our simulation results are in good agreement with the theoretical prediction.

Moreover, using the Scattering Matrix Approach we applied the extension of MM technique to determine the dispersion matrix and to analyze multiple uniaxial circular discontinuity in waveguide filled with anisotropic metamaterials. This introduced tool is applied to the modeling of large complex structures such as filters where its rapidity compared to the commercial simulation tools is verified.
